Difference between revisions of "Colloquia"

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__NOTOC__
 
__NOTOC__
  
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In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.
  
<b>UW Madison mathematics Colloquium is on Fridays at 4:00 pm. </b>
+
==September 9 , 2022, Friday at 4pm  [https://math.ou.edu/~jing/ Jing Tao] (University of Oklahoma)==
 +
(host: Dymarz, Uyanik, WIMAW)
  
<!--- in Van Vleck B239, '''unless otherwise indicated'''. --->
+
'''On surface homeomorphisms'''
  
=Fall 2021=
+
In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.
 +
==September 23, 2022, Friday at 4pm  [https://www.pabloshmerkin.org/ Pablo Shmerkin] (University of British Columbia) ==
 +
(host: Guo, Seeger)
  
== September 17, 2021, Social Sciences 5208 + [http://128.104.155.144/ClassroomStreams/socsci5208_stream.html Live Stream], [https://markshus.wixsite.com/math Mark Shusterman] (Harvard) ==
+
'''Incidences and line counting: from the discrete to the fractal setting'''
  
(hosted by Gurevich)
+
How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.
  
'''Finitely Presented Groups in Arithmetic Geometry'''
+
==September 30, 2022, Friday at 4pm [https://alejandraquintos.com/ Alejandra Quintos] (University of Wisconsin-Madison, Statistics) ==
 +
(host: Stovall)
  
I will report on recent works, in part joint with Esnault—Srinivas, and with Jarden, on the finite presentability of several (profinite) groups arising in algebraic geometry and in number theory. These results build on a cohomological criterion of Lubotzky involving Euler characteristics. I will try to explain the analogy, rooted in arithmetic topology, between these results and classical facts about fundamental groups of three-dimensional manifolds.
+
'''Dependent Stopping Times and an Application to Credit Risk Theory'''
  
== September 24, 2021, B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [https://math.wisc.edu/staff/paul-sean/ Sean Paul] (UW-Madison) ==
+
Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.
'''The Tian-Yau-Donaldson conjecture for general polarized manifolds'''
 
  
According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle L on a compact complex manifold X should be equivalent to an algebro-geometric "stability condition" satisfied by the pair (X,L). The cscK metrics are the critical points of Mabuchi's K-energy functional M, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff M satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability.  The speaker will give a non-technical general account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry​in the spirit of Arakelov, Faltings, and Bismut-Gillet- Soule.
+
In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.
 +
==October 7, 2022, Friday at 4pm  [https://www.daniellitt.com/ Daniel Litt] (University of Toronto)==
 +
(host: Ananth Shankar)
  
== October 1, 2021, B239 + [http://go.wisc.edu/wuas48 Live stream], [https://people.math.wisc.edu/~andreic/ Andrei Caldararu] (UW-Madison) ==
+
'''The search for special symmetries'''
'''Yet another Moonshine'''
 
  
The j-function, introduced by Felix Klein in 1879, is an essential ingredient in the study of elliptic curves. It is Z-periodic on the complex upper half-plane, so it admits a Fourier expansion. The original Monstrous Moonshine conjecture, due to McKay and Conway/Norton in the 1980s, relates the Fourier coefficients of the j-function around the cusp to dimensions of irreducible representations of the Monster simple group. It was proved by Borcherds in 1992.
+
What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.
  
In my talk I will try to give a rudimentary introduction to modular forms, explain Monstrous Moonshine, and discuss a new version of it obtained in joint work with Yunfan He and Shengyuan Huang. Our version involves studying the j-function around CM points (so-called Landau-Ginzburg points in the physics literature) and expanding with respect to a coordinate which arises naturally in string theory.
+
==October 14, 2022, Friday at 4pm  [https://math.sciences.ncsu.edu/people/asagema/ Andrew Sageman-Furnas] (North Carolina State)==
 +
(host: Mari-Beffa)
  
== October 8, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom] + live video on the 9th floor, [https://www.maths.ox.ac.uk/people/jon.chapman Jon Chapman] (University of Oxford) ==
+
'''Constructing isometric tori with the same curvatures'''
  
('''Wasow lecture'''; hosted by Thiffeault)
+
Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.
  
'''Asymptotics beyond all orders: the devil's invention?'''
+
For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.
  
"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.
+
The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.
  
The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.
+
== October 20, 2022, Thursday at 4pm, VV911  [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
 +
(host: Kurtz, Roch)
  
== October 11, 13, 15, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom],  '''[Mon, Wed, Fri 4-5pm]''', [https://www.maths.usyd.edu.au/u/geordie/ Geordie Williamson] (University of Sydney) ==
+
''Note the unusual time and room!''
  
('''Distinguished Lecture Series'''; hosted by Gurevich)
+
'''An introduction to counts-of-counts data'''
  
'''Geometric representation theory and modular representations'''
+
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size
  
Representation theory is the study of linear symmetry. We are interested in all ways in which a group can arise as the symmetries of a vector space. Representation theory is a remarkably rich subject, with deep connections to number theory, combinatorics, algebraic geometry, differential geometry, theoretical physics and beyond. This lecture series will focus on modular representations, i.e. those representations where our vector spaces are over a field of characteristic p. I will try to highlight some of the main questions in the field and why we are interested in answering them. It is remarkable how much is still unknown and how hard some of these questions are. I will explain the role played by geometric representation theory in our attempts to understand these questions. A fascinating blend of algebra, algebraic geometry, category theory and algebraic topology is informing our understanding of basic questions. Much remains to be understood!
+
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
  
== October 22, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], [https://math.berkeley.edu/people/faculty/vera-serganova Vera Serganova] (UC Berkeley) ==
+
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
  
(hosted by Gurevich/Gorin)
+
''References''
  
'''Supersymmetry and tensor categories'''
+
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
  
I will explain how representation theory of supergroups and
+
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
supergeometry are related to general theory of tensor categories,
 
present old and new results and open questions
 
in the field. We will see how universal tensor categories can be
 
constructed using supergroups and discuss analogy between super
 
representation theory and representation theory over the fields of
 
positive characteristic.
 
  
== October 29, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton University) ==
+
[3] Ewens WJ. Theoret Popul Biol, 3, 1972
  
(hosted by Wainger)
+
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
  
'''Polynomial averages and pointwise ergodic theorems on nilpotent groups'''
+
==October 21, 2022, Friday at 4pm  [https://web.ma.utexas.edu/users/ntran/ Ngoc Mai Tran] (Texas)==
 +
(host: Rodriguez)
 +
== November 7-9, 2022, [https://ai.facebook.com/people/kristin-lauter/ Kristen Lauter] (Facebook) ==
 +
Distinguished lectures
  
I will talk about some recent work on pointwise almost
+
(host: Yang).
everywhere convergence for ergodic averages along polynomial sequences
 
in nilpotent groups of step two. Our proof is based on
 
almost-orthogonality techniques that go far beyond Fourier transform
 
tools, which are not available in the non-commutative nilpotent
 
setting. In particular we develop what we call a nilpotent circle
 
method}, which allows us to adapt some the ideas of the classical
 
circle method to the setting of nilpotent groups.
 
  
== November 5, 2021, B239 + [http://go.wisc.edu/wuas48 Live stream], [https://faculty.washington.edu/jathreya/ Jayadev S. Athreya] (University of Washington) ==
+
== November 11, 2022, Friday at 4pm [http://users.cms.caltech.edu/~jtropp/ Joel Tropp] (Caltech)==
 +
This is the Annual LAA lecture. See [https://math.wisc.edu/laa-lecture/ this] for its history.
  
(hosted by Uyanik)
+
(host: Qin, Jordan)
 +
==November 18, 2022, Friday at 4pm  [TBD]==
 +
(reserved by HC. contact: Stechmann)
 +
==December 2, 2022, Friday at 4pm  [TBD]==
 +
(reserved by HC. contact: Stechmann)
 +
==December 9, 2022, Friday at 4pm  [TBD]==
 +
(reserved by HC. contact: Stechmann)
 +
== Future Colloquia ==
  
'''Surfaces and Point Processes'''
+
[[Colloquia/Fall2022|Fall 2022]]
  
We'll give several concrete examples of how to go from the geometry of surfaces to the study of point processes, following work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and others. We'll discuss how this "probabilistic" perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We'll discuss some pieces of joint work with Cheung-Masur, Margulis, and Arana-Herrera.
+
[[Colloquia/Spring2023|Spring 2023]]
  
== November 12, 2021, [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom], [https://sites.tufts.edu/kasso/ Kasso Okoudjou] (Tufts University) ==
+
== Past Colloquia ==
 
+
[[Spring 2022 Colloquiums|Spring 2022]]
(hosted by Stovall)
 
 
 
'''An exploration in analysis on fractals '''
 
 
 
Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support.  In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.
 
 
 
== November 19, 2021 , B239 + [http://go.wisc.edu/wuas48 Live stream],  [https://math.wisc.edu/staff/ai-albert/  Albert Ai](UW-Madison) ==
 
 
 
(reserved by the hiring committee)
 
 
 
''' Low regularity solution for quasilinear PDEs'''
 
 
 
In this talk, we will consider the low regularity well-posedness problem for a pair of quasilinear dispersive PDEs: the nonlinear wave equation, and the water waves equations. Two classical methods, energy estimates and Strichartz estimates, have historically yielded substantial but partial results toward advancing the low regularity theory. We will see how, using a special structure of the equations known as a normal form structure, combined with tools from harmonic and microlocal analysis, we can refine these classical methods to drastically improve the known results for low regularity well-posedness.
 
 
 
== December 1, 2021, Wednesday at 4pm in B239  + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [https://www.math.ucla.edu/~brianrl/ Brian Lawrence] (UCLA) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Integral points on moduli spaces'''
 
 
 
Mordell's conjecture, now a theorem of Faltings, states that an algebraic curve of genus at least two has only finitely many rational points.  Recent work with Venkatesh gives a new proof of Mordell's conjecture; the method gives some hope of proving finiteness results for any variety (even of higher dimension) that can be realized as a moduli space.  I'll discuss some recent results in this direction.
 
 
 
== December 3, 2021, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM] + live video in B239, [https://people.wgtn.ac.nz/martino.lupini Martino Lupini] (Victoria University of Wellington) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Borel-definable Algebraic Topology'''
 
 
 
In this talk, I will explain how ideas and methods from logic can be used to obtain refinements of classical invariants from homological algebra and algebraic topology. I will then present some applications to classification problems in topology. This is joint work with Jeffrey Bergfalk and Aristotelis Panagiotopoulos.
 
 
 
== December 6, 2021, Monday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM] + live watching in B239, [https://sites.google.com/site/michaellipnowski/ Michael Lipnowski]  (McGill) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Story about a dodecahedron'''
 
 
 
The Seifert-Weber dodecahedral space is a famous closed hyperbolic 3-manifold, one of the first to be discovered.  I'll describe some computations that I've done, together with Francesco Lin, on the dodecahedral space and some questions about small eigenvalues on hyperbolic manifolds which motivated them in the first place.  I'll also raise a question about (unlikely) intersections of geodesics on hyperbolic manifolds inspired by these computations.
 
 
 
== December 8, 2021, Wednesday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://padmask.github.io/ Padmavathi Srinivasan] (University of Georgia) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Degenerations of curves, rational points, and arithmetic topology'''
 
 
 
Number theory has a rich history of long standing open problems that are fairly easy to state, but are notoriously difficult to answer. The most famous among these is Fermat's Last Theorem, whose solution spurred the development of many technical tools in use today. The quest to find explicit methods to solve other Diophantine equations continues.
 
 
 
A recent method that has had spectacular success in finding rational points on curves that were previously out of reach is the "Quadratic Chabauty" method. The explicit implementation of the Quadratic Chabauty method is a formidable computational challenge. This talk will feature a simplification of the  Quadratic Chabauty method using geometric ideas, developed jointly with Besser and Mueller. Using ideas inspired by topology, we will outline new results (joint with Li, Litt and Salter) that show that most curves have no rational solutions at all, guided by Grothendieck's Section Conjecture. The key is to study degenerations in families of curves. The talk will close with various natural ways of measuring degenerations in families of curves (such as the conductor and the discriminant) and their interrelationship.
 
 
 
== December 10, 2021, Friday at 4pm in B239 + [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 Zoom stream], [http://www-personal.umich.edu/~apisa/ Paul Apisa] (University of Michigan) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Billiards, dynamics, and the moduli space of Riemann surfaces'''
 
 
 
The Hodge bundle is the space whose points correspond to a Riemann surface equipped with a holomorphic 1-form. This space admits a GL(2, R) action whose dynamics govern the geometry of the moduli space of Riemann surfaces, an object of central importance in geometry, algebra, and physics. I will describe work, joint with Alex Wright, that classifies roughly half of all GL(2, R) orbit closures. I will also describe applications to deceptively simple sounding problems about billiards in polygons. Along the way I will highlight connections to algebraic geometry, homogeneous dynamics, and more.
 
 
 
== December 13, 2021, Monday at 4pm in B239, [https://sites.google.com/view/nicole-looper/home?authuser=0 Nicole Looper] (Brown) ==
 
 
 
(reserved by the hiring committee)
 
 
 
== December 15, 2021, Wednesday at 4pm in B239, [https://people.seas.harvard.edu/~chr/ Chris Rycroft] (Harvard) ==
 
 
 
(reserved by the hiring committee)
 
  
'''Uncovering the rules of crumpling with a data-driven approach'''
+
[[Colloquia/Fall2021|Fall 2021]]
 
 
When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].
 
 
 
This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3].
 
 
 
[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).
 
[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).
 
[3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019).
 
 
 
== December 17, 2021, Friday at 4pm on [https://uwmadison.zoom.us/j/93283927523?pwd=S3V6Nlh4bUhYc0F5QzNabi9RMSthUT09 ZOOM],  [http://www.pdmi.ras.ru/~dchelkak/index_en.html Dmitry Chelkak] (ENS Paris) ==
 
 
 
(reserved by the hiring committee)
 
 
 
== December 20, 2021, Monday at 4pm in B239 + [http://go.wisc.edu/wuas48 Live stream], [https://sites.google.com/view/mnovackmath/home Matthew Novack] (IAS) ==
 
 
 
(reserved by the hiring committee)
 
 
 
'''Turbulent Weak Solutions of the 3D Euler Equations'''
 
 
 
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.
 
 
 
== Future ==
 
 
 
[[Colloquia/Spring2022|Spring 2022]]
 
 
 
== Past Colloquia ==
 
  
 
[[Colloquia/Spring2021|Spring 2021]]
 
[[Colloquia/Spring2021|Spring 2021]]

Latest revision as of 10:04, 3 October 2022


In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.

September 9 , 2022, Friday at 4pm Jing Tao (University of Oklahoma)

(host: Dymarz, Uyanik, WIMAW)

On surface homeomorphisms

In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.

September 23, 2022, Friday at 4pm Pablo Shmerkin (University of British Columbia)

(host: Guo, Seeger)

Incidences and line counting: from the discrete to the fractal setting

How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

September 30, 2022, Friday at 4pm Alejandra Quintos (University of Wisconsin-Madison, Statistics)

(host: Stovall)

Dependent Stopping Times and an Application to Credit Risk Theory

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.

In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.

October 7, 2022, Friday at 4pm Daniel Litt (University of Toronto)

(host: Ananth Shankar)

The search for special symmetries

What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.

October 14, 2022, Friday at 4pm Andrew Sageman-Furnas (North Carolina State)

(host: Mari-Beffa)

Constructing isometric tori with the same curvatures

Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.

For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.

The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.

October 20, 2022, Thursday at 4pm, VV911 Simon Tavaré (Columbia University)

(host: Kurtz, Roch)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C1, C2, … of species represented once, twice, … in a sample of size

N = C1 + 2 C2 + 3 C3 + ….  containing S = C1 + C2 + species; the vector C = (C1, C2, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.

References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)

October 21, 2022, Friday at 4pm Ngoc Mai Tran (Texas)

(host: Rodriguez)

November 7-9, 2022, Kristen Lauter (Facebook)

Distinguished lectures

(host: Yang).

November 11, 2022, Friday at 4pm Joel Tropp (Caltech)

This is the Annual LAA lecture. See this for its history.

(host: Qin, Jordan)

November 18, 2022, Friday at 4pm [TBD]

(reserved by HC. contact: Stechmann)

December 2, 2022, Friday at 4pm [TBD]

(reserved by HC. contact: Stechmann)

December 9, 2022, Friday at 4pm [TBD]

(reserved by HC. contact: Stechmann)

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